Topology - Help needed

Hi all,
From the past few days I am trying to read Kolmogorov's introductory real analysis, so far I have finished the first two chapters on set theory, metric space, but from past one week I am trying to read the third chapter on topology but this thing is going over my head, it seems so painful & hard :( does anyone know any good source on topology, no offence but please don't recommend me the video tutorial because it's very time consuming plus I am not good in learning through videos so I will really appreciate if you know any good text on topology.

Thanks in advance.

P.S. I am a self taught student whose goal is to study measure theory & then stochastic calculus, prior to Kolmogorov, I have studied Spivak's calculus, & paul's notes. Also if someone knows a good text on introductory probability then please do let me know.

I think Munkres is the standard topology reference, but I don't think you need to know topology in such a detail. Just a knowledge of metric spaces should be enough.

I think "Real Analysis" by Carothers should be a nice book to go through. It's about metric spaces, function spaces and measure theory. If you're comfortable with Spivak, then this book should be ok for you.

I think Munkres is the standard topology reference, but I don't think you need to know topology in such a detail. Just a knowledge of metric spaces should be enough.

I think "Real Analysis" by Carothers should be a nice book to go through. It's about metric spaces, function spaces and measure theory. If you're comfortable with Spivak, then this book should be ok for you.

Having just browsed the chapter it looks to me like more than 3/4 of that chapter is explained in a far more friendly fashion in Mendelson's insanely friendly topology book. Before I'd read (most of) that I thought it'd take me forever to actually get through Kolmogorov considering his approach to topology with compactums & non-empty intersections had me pulling my hair out (as it was one of the many books apparently defining every topological concept differently), but after Mendelson I'm now only scared of the last four chapters of Kolmogorov Try get the first edition of his book, he leaves a lot of fantastic little motivational paragraphs out of the third edition & structured it a little worse. Make sure you read Mendelson's chapter on metric spaces (skip the intro chapter if you want) as he motivates absolutely everything & basically sets up the framework with which you should approach mathematics from from now on :tongue2:
I actually can't wait to read Kolmogorov over the summer now without that headache thanks to Mendelson, really recommended (short book too).

In fact, some evidence in the form of one of the exercises from the book:

In a topological space, each of the terms open set, closed set, neighborhood, closure of a set, interior of a set, boundary of a set, may be characterized by any other one of these terms. Construct a table containing the thirty such possible definitions or theorems in which, for example, the entry in the row labelled interior and the column labelled open set is the characterization of interior in terms of open sets (Theorem 4.17), etc.

This is before he approaches compactness & connectedness, explaining everything you'll need (including the finite intersection stuff in Kolmogorov).

I can see why you're lost. The material in that book does not look like introductory analysis material at all. I highly suggest getting another book.

The point of topology is not an easy one to grasp. A way I always introduce topology is by the following two theorems:

Theorem 1: If [itex]f:[a,b]\rightarrow \mathbb{R}[/itex] is a continuous function, then [itex]f([a,b])[/itex] has a minimum and a maximum.

Theorem 2: If [itex]f:[a,b]\rightarrow \mathbb{R}[/itex] is a continuous function and if a<0<b, then there exists a [itex]c\in [a,b][/itex] such that f(c)=0.

This is all very well, but how can we generalize this?? The domain in both function is one-dimensional (i.e. subset of [itex]\mathbb{R}[/itex]). What if we want a 2-dimensional domain?? What if our domain is a sphere??
These questions ask for a general treatment. That is, we might ask ourselves: under what circumstances stay theorem 1 and theorem 2 true?? Research of this fact was being done, and the conclusions were that the domain in theorem 1 should be compact. And the domain in theorem 2 should be connected. So we get

Theorem 1: If [itex]f:X\rightarrow \mathbb{R}[/itex] is a continuous function and if X is compact, then [itex]f(X)[/itex] has a minimum and a maximum.

Theorem 2: If [itex]f:X\rightarrow \mathbb{R}[/itex] is a continuous function and if X is connected, then f(X) is connected (= an interval).

However, the definitions of compact and of connected are not at all easy. Even right now, I have still no intuition about why we define compactness the way we do. I know that it works and I know why it works, but it's quite mysterious.

micromass, i'vet tried to give an elementary proof of that theorem. what do you think of this? does it scan for a beginner who knows a real number is an infinite decimal? Or can you improve it?

Intermediate value theorem:
Assume f(0)<0<f(1). Then by looking at the values f(0), f(0.1), f(0.2),…,f(.9), f(1.0), there is some integer a1 between 0 and 9 so that
f(.a1) ≤ 0≤f(.a1+.1). If one of these values is zero we stop.
If not, then there is some integer a2 so that f(.a1a2) ≤ 0 ≤ f(.a1a2 +.01).
Continue…..
Either we find a point where f = 0 or else we find a sequence of decimals
xn = .a1a2….an, and xn+1 = = .a1a2….an + 1/10^n,
so that f(xn) < 0 < f(xn+1) for all n, and |xn – xn+1| < 1/10^n.
Since both sequences {an} and {an + 1/10^n} converge to the same decimal x, and since f(an)<0 while f(an +1/10^n) >0, it follows that f(x) = 0.

I know this isn't that great but I'm disillusioned with the abstract approaches to it in the usual books and wanted make it look easier. I.e. I use no axioms of least upper bounds etc... Can you help?

That's actually a nice proof since it's so constructive. You get an algorithm that way that describes how you can find (approximations) of the point in question. I never thought about proving it that way, I like it!

Thank you. Since you liked that, you inspire me to suggest analogous proofs of Spivak's other big theorems. Although the idea is the same, here is an attempt.

2) Every function f continuous on [0,1] is bounded there.
proof: if not then it is unbounded on some interval of form [.a1, .a1+.1],
hence also on some interval of form [.a1a2, .a1a2 +.01].
Continuing we find an infinite decimal x = .a1a2a3.... in [0,1], such that f is unbounded on every interval containing x. But if f is continuous at x, then f is bounded on some neighborhood of x. QED.

3) Claim: A continuous f takes on a maximum and a minimum on [0,1].
proof: By theorem 1 above (IVT in post #8) the set of values of f form an interval, and by theorem 2), they form a bounded interval. If that interval is not closed it has form say (c,d), but then the continuous function 1/(f-d) would be unbounded. QED.

I believe these proofs are as rigorous as the ones in Mike's book but much easier, and shorter. Of course Mike wrote his book for students who needed to learn the abstract approach as early as possible, and he did a fantastic job of even making it fun.

But my hope is to be more consistent and logical for the average calculus students trying to follow the ideas. I.e. if we tell them in a non honors class, or even an honors class that real numbers are infinite decimals, why not use that statement to give them direct proofs of the big theorems, instead of just saying "this is beyond the scope of the course", when really it isn't at all. I always loved abstraction, but I now believe after a lifetime of teaching that unnecessarily abstract presentations cause many students to just lose contact with the subject.

What do you think? (For openness, I learned the rigorous approach to real numbers via decimals from Spivak's appendix, while teaching a group of bright high schoolers, so in a way Mike also deserves credit for this approach.)

1) Intermediate value theorem:
Assume f(0)<0<f(1). Then by looking at the values f(0), f(0.1), f(0.2),…,f(.9), f(1.0), there is some integer a1 between 0 and 9 so that f(.a1) ≤ 0≤f(.a1+.1). If one of these values is zero we stop.
If not, then there is some integer a2 so that f(.a1a2) ≤ 0 ≤ f(.a1a2 +.01).
Continue…..
Either we find a point where f = 0 or else we find a sequence of decimals xn = .a1a2….an,

so that f(xn) < 0 < f(xn+ 1/10^n) for all n.

Since both sequences {xn} and {xn + 1/10^n} converge to the same decimal x in [0,1], and since f(xn)<0
while f(xn +1/10^n) >0, it follows that 0≤ f(x) ≤ 0. QED.

The point of topology is not an easy one to grasp. A way I always introduce topology is by the following two theorems:

Theorem 1: If f:[a,b]→R is a continuous function, then f([a,b]) has a minimum and a maximum.

Theorem 2: If f:[a,b]→R is a continuous function and if a<0<b, then there exists a c∈[a,b] such that f(c)=0.

This is all very well, but how can we generalize this?? The domain in both function is one-dimensional (i.e. subset of R). What if we want a 2-dimensional domain?? What if our domain is a sphere??
These questions ask for a general treatment. That is, we might ask ourselves: under what circumstances stay theorem 1 and theorem 2 true?? Research of this fact was being done, and the conclusions were that the domain in theorem 1 should be compact. And the domain in theorem 2 should be connected. So we get

Theorem 1: If f:X→R is a continuous function and if X is compact, then f(X) has a minimum and a maximum.

Theorem 2: If f:X→R is a continuous function and if X is connected, then f(X) is connected (= an interval).

However, the definitions of compact and of connected are not at all easy. Even right now, I have still no intuition about why we define compactness the way we do. I know that it works and I know why it works, but it's quite mysterious.

I think the motivation for topology is the old story about doughnuts and coffee mugs. And that's sort of what topologists proper actually study (except algebraic topologists are even dumber and can't tell the difference between things that have the same homotopy type or maybe things that are weak equivalent). Topology, from an analysis point of view is a bit different, but the way I think of it, it consists of the observation that things like function spaces also have a topology that you can study, just like surfaces or manifolds do (which roughly means they can be viewed as stretchy things). Functions spaces sometimes have a metric on them, so distance makes sense, and once you have distance, you can forget some of that structure and you have topology. And metric space topology isn't much different than topology of R^n, say. The gap in my understanding here is that I don't have very good examples of topological spaces that aren't metric spaces. I can cook up examples, but they aren't very interesting. So, if I knew more analysis, I could probably say it's useful to try to generalize topology to things that don't even have a metric.

Connectedness isn't that unintuitive of an idea.

Compactness is fairly non-intuitive. But the original compactness was sequential compactness, which isn't so bad, once you know a little real analysis. Sequential compactness DOES make sense in any topological space. So, while this isn't the best motivation ever, you can get some motivation for the concept of compactness this way. Sequential compactness is equivalent to compactness for metric spaces. Furthermore, for some constructions, compactness, in the open set formulation proves to be very useful. It's good for patching local information together to get global information. So, you learn of its importance while doing some proofs. Now, you could just define compactness to be sequential compactness, but the present definition of compactness is more general. So, it allows you to have that nice open set property to work with for your proofs, but is a weaker requirement on your space. So, if you want the most general setting in which you can do those kinds of local to global arguments, you want the present definition of compactness. But all this kind of assumes you are pretty comfortable with weird, but useful non-metrizable topological spaces, for which you'd like to say are compact, even though they happen not to be sequentially compact.

I'm a topology student who cares mainly about manifolds, so I don't really need the really ugly spaces that come up in analysis. So, basically, I suspect when you learn a lot of analysis (more than I know), you can really appreciate the motivation for point-set.

Another motivation for point-set topology would be to take quotient spaces (glue stuff together). Gluing operations are very useful in topology. Typically, the result will be metrizable, but you might not want to have to actually get your hands on the metric. If I want to think of a pacman square as a torus, I don't want to cook up some weird metric to study the topology. So, what that is hinting at is that the things that topologists are interested in aren't really captured by the metric, even though most of the spaces are metrizable.

So, the open set characterization of continuity gives you a way to define continuity without reference to a metric. So, if you think in terms of categories, if continuous maps are morphisms, then what structure are they preserving? It's easy to ask that about continuous with a continuous inverse (i.e. homeomorphism). If two metric spaces are homeomorphic, they have essentially the same open set structure. So, maybe the essential structure is the open sets, rather than the metric. So, then you decide what kind of behavior open sets ought to have, and you end up with the definition of a topology.

Once in a while, I find it useful to look at the concepts in topology in the following way.

It used to bother me that we had to express continuity with maps going backward, when algebra has most of it's morphisms respecting properties forward. In fact, there is a way to see continuous function this way, and while it's quite informal, it is very intuitive and can help when facing a difficult problem.

It is this:

Continuous functions map close points to close points, very informal, doesn't match up with any of the machinery of topolgy, but I can make it look like stuff in algebra:

f(a close to b) implies f(a) close to f(b)

And there's more,

When I say close, I mean how many open sets there are. So lots of open sets means we can separate points more, so points are farther away. In the discrete topology, all the points are very far from each other. On the other hand, the trivial topology is as cozy as it gets.

So any map from the discrete topology or to the trivial topology is continuous.

A compact set has not too many open sets. Points are close. So continuous maps take compact sets to compact sets.

Hausdorff, points are far enough apart; there is an important open mapping theorem that uses Hausdorffness.

While the machinery is definitely useful for theoretical purposes a bit remove from application, you can also see a more immediate relevancy in the following made up situation. Say you have a collection of data, and you want to process that data to suggest what to do next. But if you have a small error in the input, it would be bad if your function had discontinuity nearby, for then you could get a far-off output from the correct output.

there is a forward direction version of continuity along the lines you suggest. f is continuous if for every subset A in the domain, the image under f of the closure of A, is contained the closure of f(A).

i learned set theoretic topology from kelley in 1965 and have almost never needed any more. in fact i have usually needed less. the metric space theory there is unfortunately obscured by being buried within the less useful generalization of uniform spaces. the nets and ultrafilters there are also rather more esoteric than i would recommend for a first course.

but i recall he does make the various equivalent definitions of continuity clear.